The rate obtained in (REF ) is optimal when \(L = K\)  [1]}.
Interestingly, for \(L > K\)  an additional \(\log (\sigma ^2 / \Delta _k^2)\)  factor is required, which we believe to be necessary. The factor may be seen as a consequence of the fact that adding one false positive in a setting when \(K=0\)  (i.e. considering \(L=1\) ) only decreases the optimal partitioning objective function (REF ) by \(\mathcal {O}_{\mathbb {P}}(\sigma ^2 \log \log n)\) , while adding two false positives results in a decrease of \(\mathcal {O}_{\mathbb {P}}(\sigma ^2 \log n)\) , see¸ Lemmas REF  and REF , respectively. Hence, placing one incorrect change-point due to noise is more beneficial when there is at least one false positive, i.e. \(L > K\) .
